Solution of asymmetric traveling salesman problems combining the volume and simplex algorithms

In this paper we investigate the usage of the Lagrangian approach as a mechanism of speeding up the optimal basis identification in simplex methods. More precisely, we have combined the volume algorithm (a variation of the subgradient algorithm) with a simplex implementation like Cplex on a strong linear programming formulation of the asymmetric traveling salesman problem known by extended disaggregated flow formulation. We compare the efficiency of our combined approach on a selection of test problems from the TSPLIB collection and on randomly generated instances. Numerical results also include heuristic finding of integer solutions and results are compared against some well-known heuristic methods.